6.2 Properties of Power Series

Power series are versatile tools in calculus, allowing us to perform various operations like addition, multiplication, and differentiation. These operations help us manipulate and analyze functions represented as infinite sums, expanding our understanding of mathematical relationships.

By mastering power series properties, we gain insights into function behavior and convergence. This knowledge is crucial for solving complex problems in calculus and related fields, enabling us to work with infinite series representations of functions effectively.

Properties of Power Series

Operations on power series

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• Addition and subtraction performed by adding or subtracting corresponding coefficients of like terms
  • Resulting series has the same interval of convergence as the original series (e.g., $\sum_{n=0}^{\infty} a_nx^n + \sum_{n=0}^{\infty} b_nx^n = \sum_{n=0}^{\infty} (a_n + b_n)x^n$)
• Scalar multiplication achieved by multiplying each coefficient by the scalar value
  • Resulting series has the same interval of convergence as the original series (e.g., $c\sum_{n=0}^{\infty} a_nx^n = \sum_{n=0}^{\infty} (ca_n)x^n$)
• Uniform convergence of power series within their interval of convergence allows for term-by-term operations

Variable substitution in power series

• Replace the variable in the original series with a new expression
  • Determine the new interval of convergence based on the substitution (e.g., substituting $x$ with $x-1$ in $\sum_{n=0}^{\infty} a_nx^n$ results in $\sum_{n=0}^{\infty} a_n(x-1)^n$)
• Multiply each term in the series by a power of the variable
  • Adjust the indices of the summation accordingly (e.g., multiplying $\sum_{n=0}^{\infty} a_nx^n$ by $x^2$ results in $\sum_{n=0}^{\infty} a_nx^{n+2}$)
  • Resulting series may have a different interval of convergence

Multiplication of power series

• Cauchy product multiplies corresponding terms and sums the results
  • Coefficient of the $n$-th term in the product given by $\sum_{k=0}^{n} a_k b_{n-k}$
  • Resulting series has an interval of convergence at least as large as the intersection of the intervals of convergence of the original series (e.g., $(\sum_{n=0}^{\infty} a_nx^n)(\sum_{n=0}^{\infty} b_nx^n) = \sum_{n=0}^{\infty} (\sum_{k=0}^{n} a_k b_{n-k})x^n$)

Differentiation and integration of series

• Term-by-term differentiation performed by differentiating each term in the series individually
  • Resulting series has the same interval of convergence as the original series
  • $n$-th term of the differentiated series given by $na_nx^{n-1}$ (e.g., $\frac{d}{dx}\sum_{n=0}^{\infty} a_nx^n = \sum_{n=1}^{\infty} na_nx^{n-1}$)
• Term-by-term integration achieved by integrating each term in the series individually
  • Resulting series has an interval of convergence at least as large as the original series
  • Constant of integration typically chosen to be 0
  • $n$-th term of the integrated series given by $\frac{a_n}{n+1}x^{n+1}$ (e.g., $\int \sum_{n=0}^{\infty} a_nx^n dx = \sum_{n=0}^{\infty} \frac{a_n}{n+1}x^{n+1} + C$)

Advanced concepts in power series

• Analytic functions can be represented by power series within their domain of convergence
• Complex analysis extends the study of power series to the complex plane
• Laurent series generalize power series to include negative powers of the variable
• Abel's theorem relates the behavior of a power series at the boundary of its convergence interval to its sum function